Character and dimension formulae for general linear superalgebra (Q876318)

Since Kac initiated the study of the representation theory of Lie superalgebras thirty years ago, much progress has been made towards understanding the finite-dimensional irreducible representations. Most notably Serganova has determined the formal characters of these representations, obtained from the evaluation at \(q=-1\) of certain Kazhdan-Lusztig polynomials that she introduced for this purpose \textit{V. Serganova} [Sel. Math., New Ser. 2, No. 4, 607--651 (1996; Zbl 0881.17005)]. Subsequently \textit{J. Brundan} [J. Am. Math. Soc. 16, No. 1, 185--231 (2003; Zbl 1050.17004)] computed these polynomials via a relationship with canonical bases for the quantized enveloping algebra of type \(A_{\infty}\). The present paper considers the characters and dimensions of the finite-dimensional irreducible representations of the general linear Lie superalgebra. First, the authors re-work Brundan's algorithm to obtain an explicit closed formula for the Kazhdan-Lusztig polynomials. This is then used to prove a conjecture of [\textit{J. Van der Jeugt, J. W. B. Hughes, R. C. King, J. Thierry-Mieg}, J. Math. Phys. 31, No. 9, 2278--2304 (1990; Zbl 0725.17004)] which gives the character of a finite-dimensional irreducible representation as an infinite alternating sum of characters of Kac modules. Thirdly, this formula is re-written as a \textit{finite} alternating sum of so-called Bernstein-Leites characters. From this the authors deduce a closed formula for the dimension of any finite-dimensional irreducible representation; until now, no such closed formula applying in this generality had been known.